Figure 1

Schematic illustration of a dynamic force spectroscopy experiment. The receptor (e.g., a protein) is immobilized on the surface (e.g., mica) and the ligand (e.g., a DNA-fragment) is connected via a linker [e.g., poly(ethylene glycol) (PEG)] to the tip of an AFM cantilever which serves as a force transducer. The distance between the mica surface and tip can be controlled with a piezoelectric element. When the surface is pulled down at a constant speed $v$, monotonically increasing forces act on the ligand-receptor complex.Reuse & Permissions

Figure 2

(Color) Typical force-distance curve (only the so-called retracting part of the complete force-distance cycle is shown; direction of pulling from right to left, indicated by the two large arrows). Left of the point $r_2$, the connection between cantilever and surface has broken and the measured signal is caused by thermal fluctuations. Fitting this part of the curve with a straight line yields the “baseline” (solid green; extrapolated as dotted green line) which sets the force offset. Its intersection with the extrapolated dotted magenta line (“sensor response,” fitted by a straight line) at $r_0$ defines the point where the cantilever leaves the surface. Between $r_0$ and $r_1$ the measured force is caused by adhesion, thermal fluctuations, and unspecific interactions. From (approximately) $r_1$ on, the complex composed of a cantilever, linker, and the ligand-receptor molecules is stretched until the bond breaks at $r_2$. The part of the rupture curve between $r_1$ and $r_2$ is called the force-extension characteristic or loading and can be fitted by a second degree polynomial (solid red curve). The slope of this polynomial at extension $r_2$ is called stiffness and the corresponding difference between the fitted red curve and the green baseline at $r_2$ is defined as the rupture force (or dissociation force; indicated as a vertical arrow). The gray solid line is a copy of the green line shifted downwards. The (positive) distance between the green and gray lines is denoted as force threshold $f_{\text{min}}$. For the purpose of identifying rupture events, only force values beyond this threshold are taken into account. Inset: Example of a force-distance curve for which no rupture event would be accepted for further analysis for the following reasons: The discontinuity at point $r_2$ is neither the last discontinuity in the force distance cycle nor does the cantilever immediately “jump back” to the green baseline. The observed event at point $r_4$ might correspond to the rupture of a single bond, but the corresponding red force-distance curve exceeds the threshold $f_{\text{min}}$ from the very beginning.Reuse & Permissions

Figure 3

(Color) Rupture force data for protein-DNA interaction previously studied in [30] are plotted against the slope of the force-extension curve at the point of rupture $r_2$ (stiffness) in a 2D histogram (red, high frequency, blue, low frequency). At a pulling velocity of $v=5000\text{nm}/\text{s}$, 2317 rupture curves have been measured. After the automated analysis (see Sec. 3A) with threshold force $f_{\text{min}}=42\text{pN}$, 259 specific rupture events have been identified. The white solid line corresponds to the master curve which has been constructed out of the force-extension curves at the maximum of the stiffness against force plot. The cumulated distributions of the rupture force and of the stiffness are shown above and to the left of the 2D histogram. Inset: Illustration of raw data force-extension curves of the rupture events with $\left(f_i,f_i^{\prime }\right)$ falling into the marked region of the 2D histogram.Reuse & Permissions

Figure 4

(Color) Rupture force data from Fig. 3. (a) Histogram of the stiffness of all force-extension curves with rupture forces between 60 and 80 pN. (b) (Illustration) The master curve (red) is constructed as described in the text. The maximally allowed deviation of stiffness at the point of rupture is ${\delta }_1=0.2$ and the maximally allowed relative deviation from the master curve ${\delta }_2=0.12$. The blue thin lines are force-extension curves of single measurements (fitted by second degree polynomials) which have been accepted by the filtering procedure (see Sec. 3B). All these events have force-extension curves which lie entirely between the two green lines that indicate the allowed deviation from the master curve. (c) and (d) Histograms of the rupture forces before and after this filtering procedure, respectively.Reuse & Permissions

Figure 5

(Color online) Symbols: The functions $-v\text{ln}\mathbf{\left(}{\tilde{n}}_v\left(f\right)\mathbf{\right)}$ for the experimental rupture force data from [30]. Each point corresponds to one observed rupture event. A unique threshold force $f_{\text{min}}=42\text{pN}$ has been used for all pulling velocities and only events with a force-extension characteristic similar to the constructed master curve $F\left(s\right)$ have been taken into account (for more details see Sec. 3B). The number $N_v$ of experimental data points for the six different velocities are $N_{200\text{nm}/\text{s}}=54$, $N_{500\text{nm}/\text{s}}=48$, $N_{1000\text{nm}/\text{s}}=80$, $N_{2000\text{nm}/\text{s}}=151$, $N_{3000\text{nm}/\text{s}}=87$, and $N_{5000\text{nm}/\text{s}}=95$. A few large rupture forces are omitted for sake of better visibility. Solid lines: Theoretical functions for the heterogeneous bond model with force-extension characteristic $F\left(s\right)$.Reuse & Permissions

Figure 6

Same rupture force data as in Fig. 5 but with velocity dependent threshold force: $f_{\text{min}}=32\text{pN}$ for $v\le 500\text{nm}/\text{s}$, $f_{\text{min}}=38\text{pN}$ for $1000\text{nm}/\text{s}\le v\le 3000\text{nm}/\text{s}$, and $f_{\text{min}}=42\text{pN}$ for $v=5000\text{nm}/\text{s}$. The number of rupture events for each pulling velocity is indicated in brackets. (Solid lines) Maximum likelihood fit for the heterogeneous bond model. More details are given in the caption of Fig. 5. (Dashed lines) Theoretical distribution for the standard Bell model [Eqs. (1, 2, 3, 4, 5, 6)] with parameters $k_0=1.43{\text{s}}^{-1}$ and $x_{\beta }=0.52\text{nm}$ rescaled by a factor 1/3. The force-extension master curve $F\left(s\right)$ has been used for the calculation of the rupture probability density according to Eqs. (4, 5, 6). The parameters have been estimated from the full (unfiltered) raw data set in [30] according to the standard Bell model under the assumption of a linear force extension characteristic.Reuse & Permissions